PRICES OF BARRIER AND FIRST-TOUCH DIGITAL OPTIONS IN LÉVY-DRIVEN MODELS, NEAR BARRIER

We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.